Pi comes up in the most unexpected places. Here's an application to walk counting that involves it.

Suppose you have a chain of length n. How many walks of length k are there on the chain? For instance, for a chain of length 5, there are 5 paths of length 0 (start at each vertex and don't go anywhere), 8 of length 1 (traverse each edge either left-right or right-left), 14 of length 2, etc.

Curiously enough, there's an explicit formula for this

<img src="http://yaroslavvb.com/research/ideas/walk-counting/cool-formula.PNG">

So to find the number of walks of length 3 in a chain of length 5, plug n=5, k=3 into formula above, and get 

<img src="http://yaroslavvb.com/research/ideas/walk-counting/cool-formula2.PNG">

Which is 14.